We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to ``high-temperature'' problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. The proof exploits the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold rank), which via subspace enumeration lets us enumerate small cuts efficiently.

翻译：我们提出一种随机化算法，可近似计算稠密正则二分图中的独立集数量——在近似计数术语中，本文为稠密正则二分图类上的#BIS问题给出了一个全多项式随机近似方案（FPRAS）。高效计数算法通常适用于有界度图上的“高温”问题，而我们的贡献是一个显著例外，因为它适用于低温环境下的稠密图。我们的方法为组合优化领域的一系列长期研究提供了以计数为核心的补充，这些研究通过谱论证表明Max-Cut和Unique Games等约束满足问题（CSP）在稠密图上易于求解。证明利用了以下事实：稠密正则图具有某种小集扩张性质（即有限阈值秩），这一性质通过子空间枚举使我们能高效枚举小割。